Optimal. Leaf size=126 \[ -\frac {19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac {9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d} \]
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Rubi [A] time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2759, 2751, 2652, 2651} \[ -\frac {19 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{10\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{8 a d}+\frac {9 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 d} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rule 2759
Rubi steps
\begin {align*} \int \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac {3 \int \left (\frac {5 a}{3}-a \sin (c+d x)\right ) (a+a \sin (c+d x))^{2/3} \, dx}{8 a}\\ &=\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac {19}{40} \int (a+a \sin (c+d x))^{2/3} \, dx\\ &=\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}+\frac {\left (19 (a+a \sin (c+d x))^{2/3}\right ) \int (1+\sin (c+d x))^{2/3} \, dx}{40 (1+\sin (c+d x))^{2/3}}\\ &=\frac {9 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 d}-\frac {19 \cos (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{10\ 2^{5/6} d (1+\sin (c+d x))^{7/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{8 a d}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 151, normalized size = 1.20 \[ \frac {3 (a (\sin (c+d x)+1))^{2/3} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (19 \sqrt {2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )+\sqrt {1-\sin (c+d x)} (5 \cos (2 (c+d x))-14 (\sin (c+d x)+2))\right )}{80 d \sqrt {1-\sin (c+d x)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sin \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \sin ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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